Both schemes and manifolds are local ringed spaces which are locally isomorphic to spaces in some full subcategory of local ringed spaces (local models). Now, there is the inherent notion of the Zariski tangent space in a point (dual of maximal ideal modulo its square) which is the "right" definition for schemes and for $C^\infty$-manifolds (over $\mathbb{R}$ and $\mathbb{C}$). But for $C^r$-manifolds over $\mathbb{R}$ with $r<\infty$ this is not the correct definition. Here one has to take equivalence classes of $C^r$-curves through the point. Isn't there some general definition of tangent spaces which is always the right one?

I am also not completely sure what "right" means. So far, I think that one wants the dimension of the tangent space to be equal to the dimension of the point. This is for example the problem with the Zariski tangent spaces for $C^r$-manifolds. Can this failure be explained geometrically?

Wait, why is m/m^2 not the right definition for C^r manifolds?
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Kevin H. LinNov 5 '09 at 0:40

1

Because the proof that the dual of m/m^2 is isomorphic to the classical tangent space defined using differentials relies directly on the smoothness assumption if you've seen the proof.
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Harry GindiDec 9 '09 at 11:11

Another comment since I don't know enough about this to give you a reference. I was just talking to my professor today about this, and he mentioned that there's a definition using cohomology that reduces to the zariski cotangent space on nice manifolds and schemes, that is, the cohomology groups are trivial for n>N for some fixed natural number N. I don't know enough about it to tell you about it, but I hope someone will pick this up from here.
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Harry GindiDec 9 '09 at 11:18

6 Answers
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Consider the real line $\mathbb R$ and $C^1\_0$ , the ring of germs of continuously differentiable functions at zero.
Now take the ideal $M$ of germs vanishing at zero. The Zariski cotangent space $M/M^2$ has dimension the continuum (because the classes of
$x^{1+\epsilon}$
are linearly independant in the quotient
for $0<\epsilon <1$ ).
Hence the Zariski tangent space of the real line , i.e. the dual of $M/M^2$ , has dimension
$2^{continuum}$
. Some geometers might think this is a bit large for the real line.

This result is essentially exercise 13 of Chapter 3 of Spivak's Differential Geometry , Volume I.

OK, here's a uniform definition which I've used when teaching differential geometry, allowing $C^r$ manifolds with corners $M$, with $0 < r \le \infty$. It gives the right notion for $r = \omega$ too, as well as complex manifolds and smooth schemes locally of finite type over a field $k$ (using the etale topology in the latter case), but below I stick to the above context for ease of typing. (It doesn't apply to more general schemes, or non-smooth complex-analytic spaces, etc., but I don't regard that as a defect.)

First a standard lemma, then a definition. For each $m$ in $M$ define the ring of germs of $C^r$-functions $\mathcal{O}_ m$ as usual, and (by the usual integration argument) check that for a local coordinate system $(x_i)$ near $m$ every $f \in \mathcal{O}_m$ satisfying $f(m) = 0$ necessarily satisfies
$f = \sum (x_i - x_i(m))h_i$ for some $C^{r-1}$ germs $h_i$ near $m$. The Lemma is that the condition all $h_i$ vanish at $m$ is independent of the expression chosen and of the coordinate system (e.g., prove $h_i(m) = (\partial_{x_i}|_ m)(f)$). We then say $f$
vanishes to first order at $m$.

Since the physical intuition is that a tangent vector is identified with its directional derivative operator on functions near the point, and so should kill functions vanishing to first order, it is very reasonable to define the tangent space ${\rm{T}}_ m(M)$ to be the vector space of $\mathbf{R}$-linear maps $D:\mathcal{O}_m \rightarrow \mathbf{R}$ which satisfy the "Leibnitz rule at $m$" and kill all germs vanishing to first order. Then for any $(x_i)$ the operators $\partial_{x_i}|_m$ are checked to be an $\mathbf{R}$-basis, one proves a relation with velocity vectors to parametric $C^r$-curves through $m$ (recovering that important visualization), etc.

I am not sure, but I think that Anders Kock (and Bill Lawvere) might have something to say about this. Ander's Kock has developed an topos theoretic approach to Differential Geometry which includes "infinitesimal objects". Here is a quote from Kock's book Synthetic Differential Geometry:

This definition is related to one of the classical ones, where a tangent
vector at x
∈ M (M a manifold) is an equivalence class of “short paths”
t : (
−ϵ, ϵ) → M with t(0) = x. Each representative t : (−ϵ, ϵ) → M
contains redundant information, whereas our D is so small that a t :
D
→ M gives a tangent vector with no redundant information; thus,
here, tangent vectors are inﬁnitesimal paths, of “length” D.
This is a special case of the feature of synthetic differential geometry
that the jet notion becomes representable.

Since Kock's theory of differential geometry is cartesian closed, the tangent bundle of a space M is just the object M^D.

For an analogue of the object D in Algebraic Geometry, you could take Spec(k(x)/(x^2)), where $k$ is some field. Then a morphism from Spec(k(x)/(x^2)) to a scheme X is equivalent to a point x of X rational over k and an element of the zariski tangent space to x.

So although this approach to differential geometry is nonstandard, there is at least one perspective in which the idea of tangent space is unified: they are all instances of collections of maps from some "infinitesimal interval" object into your space.

Tangent vectors in a C^r-manifold are defined by mappings of an open interval into the manifold. We would like to do a similar thing in the algebraic context.

Thus we want to consider maps from the line A^1 into a scheme X. As we are considering tangent vectors at a fixed point, we may assume that the map F: A^1 -> X maps the origin to the given point of $X$.

Of course, this is not quite the right thing as it should be enough to map an open neighborhood of the origin in A^1 to $X$. The Zariski topology is (contrary to the analytic topology of manifolds) rather coarse, so let us consider étale open neighborhoods T of the origin A^1, which we map to $X$. A tangent vector to X is then an equivalence class of morphisms from pointed étale neighborhoods T of the origin of A^1 to X where we define the equivalence of two such morphisms as in the C^r-case, namely when (after restricting to a common étale neighborhood $U$) the two morphisms $f, g\colon U \to X$ have the property that df and dg coincide in the base point of $U$.

This gives the right notion at least in the case of smooth schemes of finite type over a field (non-smooth schemes definitely compare badly to manifolds): Every such scheme possesses (at least Zariski-locally) an étale map to the affine space A^n_k. This allows one to show that the above definition gives the right tangent space, namely an n-dimensional one.

P.S.: Trying to go the other way round by using synthetic differential geometry in order to imitate the algebraic definition in the context of differential geometry does not seem to be a solution as functions in SDG are always of class $\mathcal C^\infty$.

Steven already explained a bit about the key to the answer: Synthetic differential geometry (cf. nLab where the hints on a higher categorical analogue are also present), but I would like to put it in a much broader perspective, though more in words and references than really explaining, mainly due to space, time and expertise limits.

While there are mentions, in several of the answers above, of the (possibly relative) module of Kähler differentials leading to the algebraic version of cotangent space used in algebraic and analytic geometry; this partial notion predates a little bit the more fundamental work of Grothendieck, who invented an inherent geometrical way to found a differential calculus in geometry. Similarly to the differentiation in topological vector spaces, the basic idea is to approximate the maps with linear maps, but this time Grothendieck considered maps among sheaves of $\mathcal{O}$-modules over schemes; he described the linearization in the language of operations on sheaves in terms of infinitesimal neighborhoods of the diagonal $\Delta\subset X\times X$; these are described in terms of nilpotent elements in the structure sheaf; one can also define the related notion of infinitesimally close generalized points; the infinitesimal neighborhoods build up an increasing filtration, which induces a dual filtration on the hom-spaces, so called differential filtration. The union of the differential filtration is the differential part of the hom-bimodule, and its elements are regular differential operators. A "crystaline" variant of the picture related to divided powers leads to appropriate treatment of differential calculus in positive characteristics. The notion of crystal of quasicoherent sheaves is bases on the notion of infinitesimally closed generalized points; the geometric picture with pullbacks of sheaves, leads to a definition as sort of descent data, cf.

These are a dual point of view on D-modules. The descent data in abelian context are equivalent to certain formally defined connection operator, called Grothendieck connection in this case. There are now abstract versions of the algebraic correspondence between descent data in abelian context and flat Kozsul connections for the associated "Amitsur" complexes (work of Roiter, T. Brzeziński and others, cf. connection for a coring). On the other hand, Grothendieck immediately came up with nonlinear version of crystals (crystals of schemes) which are dual point of view on what some now call D-schemes.

Grothendieck's point of view on differential calculus has been soon after the discovery at the end of 1950s, introduced in works of Malgrange, Kodaira and Spencer in the development of obstruction and deformation theory for differential equations. Both works together in late 1960s motivated Lawvere, Kock and Dubuc to extend that geoemtric approach to differential calculus into differential geometry. Dubuc introduced $C^\infty$-schemes as yet another approach to manifolds, in the spirit of the theory of schemes. Lawvere did not look only at then recent work of Grothendieck (and Malgrange, Spencer...), but also at classical work on "synthetic geometry". This is a terminology which requires caution: in 19th and early 20th century, synthetic was viewed as differing from coordinatized, analytic, and pertained to either work from axioms, not referring to coordinate and even metric aspects, and some people in axiomatic descriptive geometry refer to their geometry as synthetic even now in that "clean", but less powerful sense. Another sense is that it is close to the engineering point of view that the path of a particle can be considered either as a point in the space of paths or as a map from interval into the space, what implies that the infinite-dimensional spaces of paths should exist and one should have the exponential law, i.e. we need to embed our category of spaces into closed monoidal category; there are many such embeddings of the category of manifolds available now, and some models of them offer the model $D$ of infinitesimals, which represents the functor of taking the tangent space in particular. This model has been shaped with having in mind the Grothendieck's field of dual numbers in algebraic geometry, but the language and multiplicity of models made it very flexible in the approach of the synthetic differential geometry of Kock and Lawvere. First of all they had an independent axiomatic approach as well as study of the topos theoretic models; including the study of Cahiers topos which is even more faithful to the Grothendieck's point of view. In all these models, they had nilpotent infinitesimals, like in scheme theory, but not like infinitesimals in nonstandard analysis. More recent approach of Moerdijk-Reyes offered both nilpotent and non-nilpotent infinitesimals (though possible variant related to nonstandard analysis is not known to me). For a differential geometer there are many attractive tools in synthetic differential geometry like infinitesimal simplices, enabling intuitive and effective of many quantities involving differential forms and geometry.

On the other hand, the usual differential geometry is faithfully embedded into synthetic models, so one is bound to be conservative, i.e. not to get results about usual notions in manifolds theory which are inconsistent with the usual definitions. One just gets more intuitive and technical power.

I should also mention that the Grothendieck's picture with the infinitesimal thickenings, aka resolutions of diagonals, leading to differential calculus, can be extended to noncommutative spaces rerpesented by abelian categories "of quasicoherent modules". This has been done in 1996 preprints

The resulting definition of the rings of regular differential operators on noncommutative rings has been used in the study toward the Beilinson-Bernstein correspondence for quantum groups in later published two articles, which however skip the geometric derivation of the definition of the differential operators used:

The philosophy I subscribe to: tangent vector is an object defined by a base point and a direction.

When you want to make a formal definition out of it you do the following things:

for schemes, you define tangent vector as a map from $\text{Spec}\\,\Bbbk[\epsilon]$;

for Cr-manifolds, you define tangent vector as a map from a neighborhood of 0 up to $\sim$

Is it possible to make a definition based on neighborhoods for schemes? Yes, you just have to do it technically right and consider formal neighborhood (which is the same for any smooth point of dimension n) rather then Zariski neighborhood.

Similarly, is it possible to define tangent vectors for Cr-manifolds using ideals? Yes, simple take Cr-differentiable maps modulo Cr-maps with 0 differential. Those are proper analogues of $\mathfrak m$ and $\mathfrak m^2$ in this context. The second set, for a curve, can be described as set of Cr-1-maps multiplied by x, but this is just one of the possible presentations.

Conclusion: when you look at it this way, the definitions are really similar on a deep level, but may be presented differently because of the technical details peculiar to each of the two cases.